The topology of simply-connected algebraic surfaces and other smooth 4-manifolds will be studied. Questions such as the following will be addressed: How do algebraic surfaces behave under connected sum? Which 4-manifolds are sums of algebraic surfaces? How can Donaldson's invariants be defined topologically? These are all questions at the forefront of research in four-dimensional manifold theory. A manifold is a very natural geometric object, being locally like Euclidean space, although globally more highly connected. Since the space of general reativity theory is a four-dimensional manifold, questions such as these may even have cosmological significance.
